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清华大学学报(自然科学版)  2015, Vol. 55 Issue (1): 134-140    
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Ma数预处理间断Galerkin算法
谭勤学,任静(),蒋洪德
Preconditioning discontinuous Galerkin method for low Mach number flows
Qinxue TAN,Jing REN(),Hongde JIANG
Gas Turbine Research Center, Department of Thermal Engineering,Tsinghua University, Beijing 100084, China
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摘要 

该文为研究间断Galerkin方法对低Ma数流动计算的实用性,将有限体积预处理矩阵方法引入间断Galerkin框架,针对低Ma数问题发展了三维粘性流动求解方法。行波算例(traveling wave)表明: 在间断Galerkin框架下引入预处理矩阵方法可用于低Ma数粘性流动的计算,且能保持间断Galerkin方法原有的离散精度。顶盖驱动流动、层流边界层、后台阶湍流流动和方腔内自然对流4个经典算例,检验了预处理间断Galerkin方法求解低Ma数流动的可行性及程序的可靠性。不同Ma数绕NACA0012无粘流动算例进一步表明,该文所用预处理间断Galerkin方法计算收敛速度几乎与Ma数无关。

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关键词 间断Galerkin方法预处理矩阵方法低Ma数无积分矩阵运算    
Abstract

The finite volume preconditioning method is applied to the discontinuous Galerkin method for three-dimensional viscid low Mach number flows. The traveling wave case is used to verify that the preconditioning discontinuous Galerkin method is suitable for viscid low Mach number flows and that the method retains the original discrete accuracy of the discontinuous Galerkin method. The results for four classical cases (lid driven incompressible flow, a Blasius boundary layer, a backward facing step with turbulent flow and natural convection in a square enclosure) validate the applicability of the preconditioning discontinuous Galerkin method and the reliability of the program. The flow around a NACA0012 airfoil for three different Mach numbers show that the convergence speed is independent of the Mach number.

Key wordsdiscontinuous Galerkin method    precondition method    low Mach number    Quadrature free    matrix operation
收稿日期: 2013-02-25      出版日期: 2015-05-15
引用本文:   
谭勤学,任静,蒋洪德. 低Ma数预处理间断Galerkin算法[J]. 清华大学学报(自然科学版), 2015, 55(1): 134-140.
Qinxue TAN,Jing REN,Hongde JIANG. Preconditioning discontinuous Galerkin method for low Mach number flows. Journal of Tsinghua University(Science and Technology), 2015, 55(1): 134-140.
链接本文:  
http://jst.tsinghuajournals.com/CN/  或          http://jst.tsinghuajournals.com/CN/Y2015/V55/I1/134
  理论压力p分布与计算结果对比
  理论压力x方向速度u分布与计算结果对比
  理论压力y方向速度v分布与计算结果对比
变量 网格尺度h 精度n
1/9 1/16 1/25 1/36
u 0.059 01 0.015 39 0.005 43 0.002 99 2.182
v 0.059 01 0.015 39 0.005 43 0.002 99 2.182
p 0.086 36 0.025 41 0.011 89 0.004 07 2.141
  一阶插值多项式L2误差分析
  封闭方腔内流线图
  无量纲U,V速度分布与参考值比较[10]
  边界层内速度分布与经典Blasius比较
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  x方向速度分布与实验对比图
  自然对流不同Ra数无量纲温度Θ分布
  自然对流不同Ra数无量纲x方向速度U/URef分布
  自然对流不同Ra数无量纲y方向速度V/VRef分布
Ra 不同变量误差ε/%
Umax Wmax Nuaver Numax Numin
103 -1.64 -1.69 -0.13 -0.10 -0.01
104 -1.08 -1.79 -0.04 -0.06 -0.32
105 -1.14 -1.67 -0.01 -0.10 -0.27
106 -1.12 -1.50 0.30 -2.19 -1.06
  间断Galerkin误差分析
  不同Ma数远场来流计算所得相对Ma数分布
  不同Ma数压力方程收敛历史对比
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